polyformfandomcom-20200213-history
User talk:Nekura Ca
Welcome! Hey there! We're excited to have Polyform Wiki as part of the Wikia community! There's still a lot to do, so here are some helpful tips and links to get your wiki going: *Not sure where to begin? Stop by Founder & Admin Central and check out the Blog for tips on how to jump start your wiki and make it grow! *Visit Community Central to make friends via chat, learn about new features and get updated on Wikia news and upcoming features on the Staff Blog. *Take a look at our webinar series -- where you can sign up to interact with Wikia staff, as well as watch past sessions *Be sure to check out to see what features you can enable on your wiki! *Explore our forums on Founder and Admin Central to see what other wiki admins are asking. *Lastly, visit our Help Pages to answer any specific question you may have. All of the above links are a great place to start exploring Wikia. If you get stuck or have a question you can't find the answer to -- please contact us . But most importantly, have fun! :) Happy editing! -- Sannse Hi! I'd like to mention a system of "abstract polyforms" I developed, mainly as a way to transform polyforms from pictures into mathematical representation. The program I use to solve polyform tilings uses this representation. Abstract polyform's main feature is that it's always a tree. It cannot contain loops. Polyform with loops can be still represented in this notation, but the representation might not be unique. Abstract polyform has two basic variables: polygon size p'' and polygon number ''n. First, if n'' = 1, a 1-poly-''p-form is a sequence 1,1,...,1,1 that contains p'' 1's. Each of those 1's represents one vertex of the polygon. Now, starting from any sequence, we can add a new polygon in the following way: 1. Choose any two adjacent numbers in the sequence. The sequence is cyclical (last number is followed by first), so you can choose the first and last number as well -- just move one of them to the other end of the sequence. 2. if those numbers are ''a and b'', replace the subsequence ''a,b'' with ''a+1,1,1,...1,1,b''+1. The number of 1's inserted here is ''p-2. So, if we start with a monomino 1,1,1,1, and replace first two 1's with sequence 2,1,1,2, we'll get 2,1,1,2,1,1. Now, since the sequences are cyclical, we can shift them. We can also mirror them. All this means that there is always a representation of the sequence that is lexicographically smallest. I call this the canonical representation. So, canonical representation of domino is 1,1,2,1,1,2. From this, we can derive two canonical forms of trominos: 1,1,2,2,1,1,2,2 and 1,1,2,1,2,1,1,3. If we extend this to tetrominos, we'll end up with five solutions, one of which is this one: 1,1,2,1,2,1,2,1,1,4 The polyforms are abstract -- they are pure mathematical constructs, not tied to any particular geometrical representation. If you try to show this tetromino in standard Euclidean grid {4,4}, you can use this operation: If a polyform whose largest number in sequence is q'' needs to fit into a regular {''p,q''} tiling, you can erase the ''q number and replace the two numbers adjacent to it with a single number: their sum. This operation is performed as many times as necessary until all instances of q'' disappear from the sequence. So, 1,1,2,1,2,1,2,1,1,4 gets contracted to 1,2,1,2,1,2,1,2. Of course, if a number ''greater than q'' exists in the sequence or appears during the contraction process, then the polyform cannot fit in that grid at all. This procedure can provide a unique representation polyforms that have internal vertices, but not for polyforms that contain holes. Those might end up with several distinct representations. However, as my primary goal was to generate polyform tilings, and polyforms with holes generally don't tile, I'm not too concerned about it. An important note: if a polyform's sequence turns out to be periodical, then a half (or third, fourth, etc.) of the full sequence can be also used as a polyform. This leads to tilings with enforced central symmetry of the tiles. I ''think that in Euclidean geometry any such tiling will be also found in search that uses the whole sequence. However, I know that this is not true in the hyperbolic geometry -- early in my studies I have encountered an onerous hexiamond in the {3,7} tiling that has a single solution when half of its sequence is used, and none at all when you use the full sequence. Once you have the representation of individual polyforms (which is also a way to create automatic enumeration), the rest is just the algorithm defining individual edges of each polyform and matching them against other edges. Fulgur14 (talk) 18:13, November 18, 2019 (UTC)